4.8.3 Piston Motion According to the Law u = at
n ; n > 0.
When a piston moves according to the law u ¼ at
n we found in Chap. 2 that the time
and place of formation of the shock wave are given by
t shock ¼
2c 0
a
1
n
1
γ þ 1
γ
n þ 1
n À 1
þ 1
h
i nÀ1
n :
and
x shock ¼ 2c 0
2c 0
a
1
n
γ
γ þ 1
þ
n À 1
n þ 1
!
1
n À 1
ð
Þ
nÀ1
n γ À 1 þ n γ þ 1
ð
Þ
½
Š
1
n
,
respectively. In the case of the linear ramp (n ¼ 1) where the piston moves with
uniform accelerated velocity we found that the shock wave forms at the forward front
of the wave whereas in this case the shock wave forms at some intermediate point. So
Fig. 4.14 Particle velocity is shown as a function of position at the times indicated for the
uniformly accelerated piston. The following parameters apply: γ ¼ 1.4, κ ¼ 1.2, Δx ¼ 0.3 and
Δt ¼ 0.05 (see text)
4.8 Numerical Examples of Plane Shocks
159